Firefox browser isn't like Chrome that has autologin feature. Once you close Firefox browser it logs-out of sites unlike Chrome that keep logins static. Only way is to keep the current browser open and not close it to keep the login status. But if you restart or close browser it logs out that's my experience with Firefox but this means you have to set "Remember" so you just choose the site and login and pswd to log back in. But if you don't do this you will have to remember the Login/Pswd to the site again.

The browser makes me sign in every time I open it. How do I stay signed in on this computer?

In Settings > Privacy & Security > make sure this is not checked > Delete cookies and site data when Firefox is closed.